Induced disjoint paths in claw-free graphs

  • Authors:

  • Petr A. Golovach;Daniël Paulusma;Erik Jan van Leeuwen

  • Affiliations:

  • School of Engineering and Computer Science, Durham University, England;School of Engineering and Computer Science, Durham University, England;Dept. Computer, Control, Managm. Eng., Sapienza University of Rome, Italy

  • Venue:
  • ESA'12 Proceedings of the 20th Annual European conference on Algorithms
  • Year:
  • 2012

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Abstract

Paths P1,…,Pk in a graph G=(V,E) are said to be mutually induced if for any 1≤ij≤k, Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test whether a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that Pi connects si and ti for i=1,…,k. This problem is known to be NP-complete already for k=2, but for n-vertex claw-free graphs, Fiala et al.gave an nO(k)-time algorithm. We improve the latter result by showing that the problem is fixed-parameter tractable for claw-free graphs when parameterized by k. Several related problems, such as the k-in-a-Path problem, are shown to be fixed-parameter tractable for claw-free graphs as well. We prove that an improvement of these results in certain directions is unlikely, for example by noting that the Induced Disjoint Paths problem cannot have a polynomial kernel for line graphs (a type of claw-free graphs), unless NP ⊆ coNP/poly. Moreover, the problem becomes NP-complete, even when k=2, for the more general class of K1,4-free graphs. Finally, we show that the nO(k)-time algorithm of Fiala et al.for testing whether a claw-free graph contains some k-vertex graph H as a topological induced minor is essentially optimal by proving that this problem is W[1]-hard even if G and H are line graphs.